Problem: A circle has a radius of $6$. An arc in this circle has a central angle of $\dfrac{19}{12}\pi$ radians. What is the length of the arc? ${12\pi}$ ${\dfrac{19}{12}\pi}$ $\color{#DF0030}{\dfrac{19}{2}\pi}$ ${6}$
First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (6) = 12\pi$ The ratio between the arc's central angle $\theta$ and $2 \pi$ radians is equal to the the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{2 \pi} = \dfrac{s}{c}$ $\dfrac{19}{12}\pi \div 2 \pi = \dfrac{s}{12\pi}$ $\dfrac{19}{24} = \dfrac{s}{12\pi}$ $\dfrac{19}{24} \times 12\pi = s$ $\dfrac{19}{2}\pi = s$